Question

如圖16，在平面直角坐標系中，直線與軸交于點，與軸交于點，拋物線經(jīng)過三點．

（1）求過三點拋物線的解析式并求出頂點的坐標；

（2）在拋物線上是否存在點，使為直角三角形，若存在，直接寫出點坐標；若不存在，請說明理由；

（3）試探究在直線上是否存在一點，使得的周長最小，若存在，求出點的坐標；若不存在，請說明理由．

Answer 1

解：（1）直線與軸交于點，與軸交于點．

，························································································· 1分

點都在拋物線上，

  

拋物線的解析式為························································ 3分

頂點······························································································· 4分

（2）存在··············································································································· 5分

············································································································· 7分

············································································································ 9分

（3）存在·············································································································· 10分

理由：

解法一：

延長到點，使，連接交直線于點，則點就是所求的點．

                       ····················································································· 11分

過點作于點．

點在拋物線上，

在中，，

，，

在中，，

，，··············································· 12分

設(shè)直線的解析式為

   解得

································································································ 13分

   解得 

在直線上存在點，使得的周長最小，此時．··· 14分